This lecture establishes the linear-algebraic and statistical foundations of the Wiener FIR filter within adaptive signal processing theory. It builds on the properties of Hermitian and positive (semi-)definite matrices to prove that any input autocorrelation matrix R is Hermitian and positive semi-definite in general, and strictly positive definite when the underlying random process is "full rank" (i.e., no sample is expressible as an exact linear combination of the others). Using this positive-definiteness, the lecture derives the Wiener-Hopf solution for the optimal FIR filter weights, w_opt = R⁻¹p, by minimizing the mean square error between the filter output and a jointly stationary target signal via matrix calculus (gradient of quadratic and linear forms with respect to a weight vector).
This lecture establishes the linear-algebraic and statistical foundations of the Wiener FIR filter within adaptive signal processing theory. It builds on the properties of Hermitian and positive (sem…